Chapter 2 - Foundations of Ray Optics

Abstract

Ray optics, or geometrical optics, is based on the short-wavelength approximation of electromagnetic theory. It is defined in terms of a package of rules (the rules of geometrical optics) that can be arrived at from the Maxwell equations in a consistent approximation scheme, referred to as the eikonal approximation, which is briefly outlined in this chapter. The basic concepts of geometrical wave front, the ray path, and the optical path length are elaborated, and the rules relating to the transport of the field vectors along ray paths are explained. The laws of reflection and refraction are derived, and the approximation scheme leading to the Fresnel formulae is briefly outlined. Basic ideas in variational calculus are explained. The characterization of ray paths in terms of Fermat’s principle is elaborated, and the nature of stationarity of the optical path length is explained with examples. The Lagrangian and Hamiltonian formulations of geometrical optics are outlined, where the determination of the ray path is seen to be analogous to a constrained problem in mechanics. The concepts of caustics and conjugate points are explained, with examples. The path integral formulation is outlined as a useful heuristic principle in geometrical optics. The Luneburg-Kline approach, which gives geometrical optics a secure foundation, is briefly mentioned.